Advanced handouts

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Co-authored-by: Mark <mark@betalupi.com>
Co-committed-by: Mark <mark@betalupi.com>

src/Advanced/Generating Functions/parts/00 introduction.tex

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+\section{Introduction}
+
+\definition{}
+Say we have a sequence $a_0, a_1, a_2, ...$. \par
+The \textit{generating function} of this sequence is defined as follows:
+\begin{equation*}
+	A(x) = \sum_{n=0}^\infty a_nx^n = a_0 + a_1x + a_2x^2 + a_3x^3 + ...
+\end{equation*}
+
+Under some circumstances, this sum does not converge, and thus $A(x)$ is undefined. \par
+However, we can still manipulate this infinite sum to get useful results even if $A(x)$
+diverges.
+
+\problem{}
+Let $A(x)$ be the generating function of the sequence $a_n$, \par
+and let $B(x)$ be the generating function of the sequence $b_n$. \par
+Find the sequences that correspond to the following generating functions:
+\begin{itemize}[itemsep=2mm]
+	\item $cA(x)$
+	\item $xA(x)$
+	\item $A(x) + B(x)$
+	\item $A(x)B(x)$
+\end{itemize}
+
+\begin{solution}
+	\begin{itemize}[itemsep=2mm]
+		\item $cA(x)$ corresponds to $ca_n$
+		\item $xA(x)$ corresponds to $0, a_0, a_1, ...$
+		\item $A(x) + B(x)$ corresponds to $a_n+b_n$
+		\item $A(x)B(x)$ is $a_0b_0 + (a_0b_1 + a_1b_0)x + (a_0b_2 + a_1b_1 + a_2b_0)x^2 + ...$ \par
+		Which corresponds to $c_n = \sum_{k=0}^n a_kb_{n-k}$
+	\end{itemize}
+\end{solution}
+
+
+\vfill
+\pagebreak
+
+
+
+
+\problem{}<xminusone>
+Assuming $|x| < 1$, show that
+\begin{equation*}
+	\frac{1}{1-x} = 1 + x + x^2 + x^3 + ...
+\end{equation*}
+\hint{use some clever algebra. What is $x \times (1 + x + x^2 + ...)$? }
+
+\begin{solution}
+	Let $S = 1 + x + x^2 + ...$ \par
+	Then, $xS = x + x^2 + x^3 + ...$ \par
+
+	\vspace{2mm}
+
+	So, $xS = S - 1$ \par
+	and $1 = S - xS = S(1 - x)$ \par
+	and $S = \frac{1}{1-x}$.
+\end{solution}
+
+
+\vfill
+
+
+
+
+\problem{}
+Let $A(x)$ be the generating function of the sequence $a_n$. \par
+Find the sequence that corresponds to the generating function $\frac{A(x)}{1-x}$
+
+\begin{solution}
+	\begin{align*}
+		\frac{A(x)}{1-x}
+		&=~ A(x)(1 + x + x^2 + ...) \\
+		&=~ (a_0 + a_1x + a_2x^2 + ...)(1 + x + x^2 + ...)\\
+		&=~ a_0 + (a_0 + a_1)x + (a_0 + a_1 + a_2)x^2 + ...
+	\end{align*}
+
+	Which corresponds to the sequence $c_n = \sum_{k=0}^n a_k$
+\end{solution}
+
+\vfill
+
+\problem{}
+Find short expressions for the generating functions for the following sequences:
+\begin{itemize}
+	\item $1, 0, 1, 0, ...$
+	\item $1, 2, 4, 8, 16, ...$
+	\item $1, 2, 3, 4, 5, ...$
+\end{itemize}
+
+\begin{solution}
+	\begin{itemize}[itemsep=2mm]
+		\item $1, 0, 1, 0, ...$ corresponds to $1 + x^2 + x^4 + ...$. \par
+		By \ref{xminusone}, this is $\frac{1}{1-x^2}$.
+
+		\item $1, 2, 4, 8, 16, ...$ corresponds to $1 + 2x + (2x)^2 + ...$. \par
+		By \ref{xminusone}, this is $\frac{1}{1-2x}$.
+
+		\item $1, 2, 3, 4, 5, ...$ corresponds to $1 + 2x + 3x^2 + 4x^3 + ...$.\par
+		This is equal to $(1 + x + x^2 + ...)^2$, and thus is $\left(\frac{1}{1-x}\right)^2$
+	\end{itemize}
+\end{solution}
+
+
+\vfill
+
+\pagebreak